How to stop independent actions on multiple armatures from fighting with each other
Each armature has its own reference to an Action datablock.  While it is possible for each armature to refer to an identical action (because datablocks are reusable) this would usually only be used when both armature objects use the same armature data (or use compatible armature data with compatible bone names).

It is unclear what you are trying to accomplish and why this behavior is a problem for you.

Just understand that it is inaccurate to say "I have multiple armatures under one action ".  What blender can do is have multiple armature objects which use the same action.  If you want an object to act differently than another object, they need to have different action datablocks in their animation_data.action field